Isolated, slowly evolving, and dynamical trapping horizons: Geometry and mechanics from surface deformations

Abstract

We study the geometry and dynamics of both isolated and dynamical trapping horizons by considering the allowed variations of their foliating two-surfaces. This provides a common framework that may be used to consider both their possible evolutions and their deformations as well as derive the well-known flux laws. Using this framework, we unify much of what is already known about these objects as well as derive some new results. In particular we characterize and study the "almost-isolated" trapping horizons known as slowly evolving horizons. It is for these horizons that a dynamical first law holds and this is analogous and closely related to the Hawking-Hartle formula for event horizons. *

Black holes and black hole thermodynamics without event horizons

“…It is also worth noting that the parameters appearing in the first law, such as the mass, angular momentum and charge, are intrinsically defined on the horizon and arise due to the boundary of the Hamiltonian formed by the isolated horizon. A related version of the first law was given by Booth and Fairhurst for slowly evolving horizons [23,24] where the horizon area is allowed to increase, but only slowly.…”

“…A trapping horizon can be a spacelike, null or timelike hypersurface. The connections between the various formulations was given in a unified framework in [69] and [24].…”

Angular momentum conservation for dynamical black holes

“…To summarize this section, the definition (39) of angular momentum can be made generically unique if the axial vector is a coordinate vector, (41), and generates a symmetry of the area form, (40). The construction can be applied in any situation where Dθ ξ ∼ = 0 almost everywhere, though the physical interpretation as angular momentum seems to be safest in the case of two poles, which locate the axis of rotation.…”

“…Previous versions of angular momentum flux laws for dynamical black holes [36,37,38,39,40,41,55] contain different terms, which are not in energy-tensor form, i.e. some tensor contracted with ψ and τ .…”

“…However, the "3+1" formalism used to describe spatial trapping horizons breaks down when the horizon becomes null, so that the isolated-horizon and dynamical-horizon frameworks were essentially distinct. Some connections were drawn, particularly for slowly evolving horizons by Booth & Fairhurst [39,40,41]. Recently, Andersson et al [42] showed that a stable trapping horizon is, on any one marginal surface, either spatial or null everywhere, so that transitions between the two types happen simultaneously on a marginal surface.…”